A convergent series is a series of numbers that has a finite sum. In other words, as you add up more and more terms of the series, the sum approaches a specific value. On the other hand, an absolutely convergent series is a series in which the absolute values of the terms converge to a finite sum. In other words, the series converges when you take the absolute value of each term and then add them up. Absolutely convergent series have the property that rearranging the terms does not change the sum, while for convergent series, rearranging the terms can change the sum. **

To determine if a series is convergent, we need to analyze the behavior of its terms as the number of terms approaches infinity. If the terms of the series approach a finite value as the number of terms increases, then the series is convergent. On the other hand, if the terms do not approach a finite value, the series is divergent. **

Yes, if b is a convergent positive sequence, then the series Σb_n will also be convergent. This is because the convergence of the sequence b_n implies that the terms of the sequence approach a finite limit as n goes to infinity. As a result, the terms of the series Σb_n will also approach zero, and the series will converge. Therefore, the convergence of the sequence b_n guarantees the convergence of the series Σb_n. **

To determine whether a series is absolutely convergent, conditionally convergent, or divergent, we need to consider both the original series and the absolute value of the series. If the original series converges and the absolute value of the series also converges, then the series is absolutely convergent. If the original series converges but the absolute value of the series diverges, then the series is conditionally convergent. If the original series diverges, then the series is divergent. **

Yes, this series is convergent because it satisfies the conditions of the alternating series test. The terms of the series alternate in sign and decrease in absolute value, and the limit of the absolute value of the terms approaches zero as n approaches infinity. Therefore, by the alternating series test, the series is convergent. **

To determine if a series or sequence is convergent, we need to check if its terms approach a specific value as the number of terms increases. For a series, we can use tests such as the ratio test, root test, or comparison test to determine convergence. For a sequence, we can check if the terms approach a specific limit. If the terms of the series or sequence approach a specific value as the number of terms increases, then it is convergent. If the terms do not approach a specific value, then it is divergent. **

The proof that a rearrangement of an absolutely convergent series is also convergent lies in the fact that absolute convergence implies convergence. Since the series is absolutely convergent, we know that the sum of the absolute values of the terms converges. Therefore, no matter how we rearrange the terms, the rearranged series will still converge to the same sum as the original series. This is because the convergence of the rearranged series is guaranteed by the convergence of the absolute values of the terms. **

The series is divergent because the terms of the series do not approach a finite limit as the number of terms increases. In other words, the sum of the terms of the series does not approach a specific value as more terms are added. This can happen if the terms of the series do not decrease in magnitude or if they oscillate in a way that prevents the sum from converging to a finite value. As a result, the series does not converge and is considered divergent. **